By Eduardo J. Dubuc

ISBN-10: 3540049347

ISBN-13: 9783540049340

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Since CT ( T e T ) use again). If in- mi = (U i 0 W k ) k e K ; we see that refines T We now take the d i r e c t e d col~mit (over T ). This sends the 59 EU, U i E(u) PC ~ T TPcTnui~ E c (U) "~ ~ie~l ~, U ~a~ E Ti (U) i :" ~ ~6~ . E ~- " (U) i i ui ) ECT flU i U'b. U ~ C~ TI uc v nu i P (C,r RU i ) P CT) = 1 i~I P(Ti) P Ti) ) Fig. ) E(U) square E uC below right. ) iel I p~I PC~ iel It 1 UT > i~I E'i(Ui) E e (U) E(U) > collm-l~ I E i(U i) T where we know that u t is induced by all It follows is a monomorphism (beginning of the proof) uT, hence is also a monomorphism.

Ba-m-lbam+l, b-l(b-m+la-lbabm-1)b The latter quickly reduces to m ~ 0° But b-l(a-mbam)b = b-ma-lbab m = a-m-lbam+l. It is clear from the relations can be non-abelian. = of G that no finite quotient It can also be shown that every m e t a b e l i a n quotient of G is abelian. REFERENCES 1. S. Mac Lane, Natural associativity and commutativity, University Studies, 49, (1963), 28-46. Rice City U n i v e r s i t y of New York DIRECTED COLINITS AND SHEAVES Pierre Antoine Received We extend, zing C 3 abelian results about categories, sheaves, every morphism for some monomorphism assumed that morphism such if categories m nitary variety tegories (for instance, f = mp p ; it is furthermore f is a regular a terminology which every epi- we call is tri- finitary is a regular category; of Grothendieck we m e n t i o n e d is extended of C 3 regular the category In such a category, directed colimits Gray's condition is complete and and that as a directed colimit ~(X,C) of sheaves then ~(X,C) is coreflexive sheaf also holds then any product is required or infi- so is any terminates limits Assume regular (hence behave also that it a C 4 re- that products colimits ca- very nice- furthermore (then we call of directed to a respect can be des- of products.

The by all EC U,V through c. ] E cU , V on the 9. Let E(U) = c(U) : C. If V ~ U and and the m o r p h i s m s , > P(CAV*CNV) : Ec(U) restriction > EC~v(V) map (unique EU, V : E(U) such > E(V) is (C e ~(U)) One sees that and f u r t h e r m o r e on as follows. C) induces such that of = ( U i n V ) i e I e {(V) nuknv P (~)gc ' . One sees that does not d e p e n d h" = j,kel ~ PU j n U k , U j on equalizers unique ) E(U) > P(C nv) induced ) E from two steps should the case in a C% regular 3. The result is now a m o r p h i s m P result to a sheaf is that, colimits terminate actually of p r e s h e a v e s factors uniquely in an exact category and a p r o j e c t i v e a sufficient (MR 26 ~ 1887), tegories step : Ec(U) p.

### Kan extensions in Enriched Category Theory by Eduardo J. Dubuc

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